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B Mathematical Appendix Taking the limit, K lim K→∞ r = lim K→∞ = lim K→∞ r α/K 1 − i + α/K α/K i + α/K K! (α/i) r 1 − α/i (K − r)!r! (K + α/i) r K(K − 1) . . . (K − r) (K + α/i) r (α/i) r r! = (α/i)r exp{α/i} . r! K−r K + α/i 1 − α/i K + α/i K 1 − α/i K + α/i −r K 1 − α/i K + α/i −r (B.13) The last equality is obtained from the fact that the limit of the first and the last term is 1 and K 1 lim = exp{−x}. K→∞ 1 + x/K (B.14) Binomial Series (x + y) n = n k=0 Series expansion for the natural logarithm 124 n x k k y n−k (B.15) ln(1 + z) = z − 1 2 z2 + 1 3 z3 − . . . , |z| ≤ 1, andz = 1 (B.16)
Bibliography D. Aldous. Exchangeability and related topics. In École d’Été de Probabilités de Saint- Flour XIII–1983, pages 1–198. Springer, Berlin, 1985. C. E. Antoniak. Mixtures of Dirichlet processes with applications to Bayesian nonparametric problems. Annals of Statistics, 2:1152–1174, 1974. R. Batley and A. Daly. On the equivalence between elimination-by-aspects and generalised extreme value models of choice behaviour. Journal of Mathematical Psychology, 50:456–467, 2006. R. R. Batsell, J. C. Polking, R. D. Cramer, and C. M. Miller. Useful mathematical relationships embedded in Tversky’s elimination by aspects model. Journal of Mathematical Psychology, 47:538–544, 2003. M. J. Beal and P. Krishnamurthy. Clustering gene expression time course data with countably infinite hidden markov models. In Proceedings of UAI, volume 22, 2006. M. J. Beal, Z. Ghahramani, and C. E. Rasmussen. The infinite hidden markov model. In Z. G. T. Dietterich, S. Becker, editor, Advances in Neural Information Processing Systems, volume 15, pages 577–584. MIT Press, 2003. J. M. Bernardo and A. F. M. Smith. Bayesian Theory. Chichester: Wiley, 1994. D. Blackwell and J. B. MacQueen. Ferguson distributions via Pólya urn schemes. Annals of Statistics, 1:353–355, 1973. D. Blei, T. L. Griffiths, M. I. Jordan, and J. B. Tenenbaum. Hierarchical topic models and the nested chinese restaurant process. In S. Thrun, L. Saul, and B. Schölkopf, editors, Advances in Neural Information Processing Systems, volume 16. MIT Press, Cambridge, MA, 2004. D. M. Blei and M. I. Jordan. Variational methods for the Dirichlet process. Journal of Bayesian Analysis, 1(1):121–144, 2005. G. Box and G. C. Tiao. Bayesian Inference in Statistical Analysis. John Wiley & Sons, 2003. R. Bradley and M. Terry. The rank analysis of incomplete block designs. I. the method of paired comparisons. Biometrika, 39:324–345, 1952. C. A. Bush and S. N. MacEachern. A semiparametric Bayesian model for randomised block designs. Biometrika, 83(2):275–285, 1996. 125
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Nonparametric Bayesian Discrete Lat
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Matrizen mit unendlich vielen Spalt
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Contents Zusammenfassung iii Abstra
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List of Algorithms 1 Gibbs sampling
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Notation Matrices are capitalized a
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Symbol Meaning IBP Z binary latent
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1 Introduction belief in the prior.
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2 Nonparametric Bayesian Analysis b
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2 Nonparametric Bayesian Analysis s
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3 Dirichlet Process Mixture Models
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3.1 The Dirichlet Process the perfo
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α G o G θi x i N 3.1 The Dirichle
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15 10 5 −0.5 0 0.5 2 1 G 0 0 −0
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increment process with the correspo
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α G o π k c i θk x i 8 N 3.1 The
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3.1 The Dirichlet Process Eq. (3.21
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number of components, K number of c
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3.2 MCMC Inference in Dirichlet Pro
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and Bush and MacEachern (1996). 3.2
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3.2.2 Algorithms for non-Conjugate
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∗ π ∗ π s 3.2 MCMC Inference
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model can be written in the form of
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−1 µ y Σy Σy D ξ normal R 3.3
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the log likelihood term is: where a
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3.3 Empirical Study on the Choice o
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autocovariance coefficient 1 0.8 0.
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# of data points # of data points 5
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3.4 Dirichlet Process Mixtures of F
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µ y Σy ξ R 0 ν w normal µ −1
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ch1 ch2 ch3 ch4 3.4 Dirichlet Proce
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3.4 Dirichlet Process Mixtures of F
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# of components # of components # o
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ch 2 ch 3 ch 1 ch 2 ch 3 ch 4 3.4 D
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3.5 Discussion 3.5 Discussion In th
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4 Indian Buffet Process Models matr
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4 Indian Buffet Process Models In t
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4 Indian Buffet Process Models α z
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Page 130 and 131: 5 Conclusions has been defined as a
Page 132 and 133: A Details of Derivations for the St
Page 135 and 136: B Mathematical Appendix B.1 Dirichl
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Page 139: Construction of A Process B.4 Equal
Page 143 and 144: Bibliography T. S. Ferguson. Prior
Page 145 and 146: Bibliography L. F. James and J. W.
Page 147 and 148: Bibliography R. M. Neal. Probabilis
Page 149 and 150: Bibliography Y. W. Teh, M. I. Jorda
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